Asymmetries in Yield Curves: Some Empirical Evidence from Ghana

Transcription

1 MPRA Munich Personal RePEc Archive Asymmetries in Yield Curves: Some Empirical Evidence from Ghana Bernard Njindan Iyke Deakin University January 2017 Online at MPRA Paper No , posted 16 May :02 UTC

2 Asymmetries in Yield Curves: Some Empirical Evidence from Ghana Bernard Njindan Iyke Department of Finance Deakin Business School Deakin University 221 Burwood Highway Melbourne, Australia Victoria / Previous Version: October 2016 This Version: January

3 Asymmetries in Yield Curves: Some Empirical Evidence from Ghana Abstract We analyze the co-movements of the monetary policy rates (MPR) and the treasury bill rates (TBR) in Ghana over the period January 2007 to July 2016, using three nonlinear econometric techniques. We find the MPR and the TBR to be cointegrated with threshold adjustments. Positive deviations from the long-term equilibrium due to increases in the MPR or decreases in the TBR are corrected at 0.3% monthly. Negative deviations from the long-term equilibrium due to decreases in the MPR or increases in TBR are corrected at 8.8% monthly. Our results show bidirectional causal flow between MPR and TBR. In addition, we find positive deviations in the TBR to be corrected at 0.34% monthly, while negative deviations are corrected at 8.6% monthly, in the short term. Thus, the TBR responds faster to negative than positive deviations. These findings are broadly consistent with the inflation-targeting framework of the Bank of Ghana. Keywords: Asymmetric Adjustments, Threshold Cointegration, Nonlinear Causality, Yield Curves, Ghana JEL Classification: E43; C22 1. Introduction In the finance and economics literature, one of the most widely studied concepts is the yield curve. The yield curve is a graph that demonstrates the theoretical and empirical relationship between interest rates and their maturities (see Taylor, 1992; Malkiel, 2015). The yield curve serves useful purposes in decision-making. First, the yield curve serves as a crucial leading indicator for business cycle forecasting. Inverted yield curves are known to signal potential recessions, as noted in the literature (see Estrella and Hardouvelis, 1991; Estrella and Mishkin, 1997). Some studies even argue that yield curves offer superior forecast of the economy than large-scale computer-based models (see Harvey, 1988; 1991; 1993; Bernanke; 1990; Chen, 1991). Second, yield curves can also provide superior indicators of monetary policy than monetary growth rates (see McCallum, 1983; Bernanke and Blinder, 1989; Piazzesi, 2010). Third, most investors utilize yield curves to forecast interest rates, bond prices, and to manage investment portfolios. Aling and Hassan (2012), for example, observe that short-term interest rates, in particular, are essential in the valuation of interest rate derivatives. Despite its immense importance, the yield curve remains one of the least studied concepts in the developing countries literature. If we narrow our scope to the Ghanaian context, yield curves have barely received research attention. The only existing study on yield curves in Ghana is that of Dzigbede and Ofori (2004), which forecasts real interest rates using yield curves in linear settings. Given that Ghana is one of the developing countries that has persistent inflation, 2

4 depreciation, and excessive economic underperformance problems in recent memory, the lack of theoretical and empirical insight into crucial concepts such as the yield curves of the country poses enormous policy challenges. For instance, how do policymakers of this country predict economic downturns? How do investors and players in the financial industry evaluate assets? This paper contributes to the literature by investigating the nature of yield curves in Ghana within an asymmetric setting. In particular, we investigate whether short- and long-term interest rates are cointegrated. Then, we analyze the direction of causal influence between the interest rates. In some sense, our paper explores the expectation theory of yield curves which posits a strong relationship between short- and long-term interest rates. We restrict our sample to the period January 2007 to July This period covers the inflation-targeting regime, when the Bank of Ghana (BOG) officially introduced forward-looking monetary policies. During this period, the BOG has reacted to prevailing market conditions which includes the behaviour of treasury bill rates (TBR) by adjusting the monetary policy rate (MPR). Therefore, short- and long-term interest rates are more likely to exhibit strong co-movement under this regime than under any other regime. Moreover, the behaviour of the yield curves during the inflationtargeting regime may influence future decisions of investors and policymakers than the previous regimes. These factors make it all the more important for us to consider this study period. This paper differs from the existing studies in two ways. First, we explore both nonlinear cointegration and causality between interest rates with different maturities. Second, we do so using a data set for a developing economy. The econometric techniques that we utilize are efficient and are found to fit data on macroeconomic variables such as interest rates well. These econometric techniques are: Kapetanios et al. (2003) nonlinear unit root test; Enders and Siklos (2001) threshold cointegration test; and asymmetric error correction model with threshold adjustments documented in Granger and Lee (1989), Balke and Fomby (1997), and Enders and Granger (1998). The remaining sections of this paper are organized as follows. In the next section, we review the literature. Then, in section 3, we discuss our methodology and the data. In section 4, we report and discuss our results. In the final section, we present the concluding remarks. 2. The Literature 2.1 The Theoretical Literature The theoretical literature on yield curves the expectation theory, in particular implicates a steady-state relationship between interest rates (see, for example, Vasicek, 1977; Richard, 1978; Cox et al., 1985; Hall et al., 1992; Bekaert and Hodrick, 2001; Malkiel, 2015). On the basis of the expectation theory of yield curves, interest rates are supposed to co-move over time (see Sarno et al., 2007). The expectation theory posits that the yield on long-term financial instruments are an average of the yield of short-term financial instruments during the holding period (or the life) of the longterm financial instruments. The latent assumption is that investors are risk-neutral; hence investors will not pay a premium to lock in a long-term interest rate (see Meiselman, 1962; McFayden et al., 1991; Arize et al., 2002; Sarno et al., 2007). Given these assumptions and 3

5 supposing that long-term financial instruments are infinitely held, the expectation theory of yield curves relates interest rates on long-term financial instruments to the expected interest rates on short-term financial instruments as RR tt nn = 1 ωω ωω jj EE tt rr tt+jj jj=0 (1) where RR tt is the long-term interest rate at time tt, rr tt+jj is the short-term interest rate at time tt + jj, EE tt is the expected value notation, ωω = 1 1+RR, RR is the mean long-term interest rate, EE tt rr tt+jj is the present value of the future short-term interest rate, and jj = 0, 1,,. According to the expectation theory, the long-term and the short-term interest rates share steadystate relationships (see Shiller, 1979; MacDonald and Speight, 1988; McFayden et al., 1991). This is in contrast with the segmented market theory which contends that the long-term and the short-term interest rates are not related. That is, the demand and supply in the markets for shortterm and long-term instruments is determined independently (see Van Home, 1978; McEnally, 1983). Following the expectation theory, Eq. (1) imposes some form of cointegrating relationship between RR and rr. This hypothesized cointegrating relationship is what we test in this paper. 2.2 The Empirical Literature The cointegrating relationship between interest rates has remained under-examined in the empirical literature, judging from the important policy implications of such relationships. Yet, the available ones have conflicting conclusions on the relationship between interest rates. In their single-country studies, Stock and Watson (1988), MacDonald and Speight (1988), McFadyen et al. (1991), Hall et al. (1992), Wallace and Warner (1993), Mandeno and Giles (1995), Enders and Siklos (2001), Della Corte et al. (2008), and Dube and Zhou (2013) find cointegrating relationship between interest rates to exist. Similarly, Arize et al. (2002) find the evidence in support of the cointegrating relationship between interest rates in their multi-country study. These empirical findings lend strong support for the expectation theory of yield curves, which suggests that interest rates co-move closely over time. Other studies find no evidence of cointegrating relationship between interest rates. Such studies are Taylor (1992), Mustafa and Rahman (1995), Bekaert and Hodrick (2001), Clarida et al. (2006), and Sarno et al. (2007). The evidence of no cointegrating relationship between interest rates is in line with the segmented-market theory, which argues that interest rates are unrelated (see McFayden et al., 1991; Arize et al., 2002). In spite of the fact that the existing empirical studies are remarkably conflicting, they appear to concentrate on just developed countries. 1 As noted earlier in this paper, yield curves offer 1 The only available studies which consider a developing country are those of Arize et al. (2002), and Dube and Zhou (2013) 4

6 important policy understanding of the state of an economy. For example, should the expectation theory be supported, it will imply that central banks can exert some influence on the long-term interest rates using instruments from the short-term market (see Arize et al., 2002). That is, the central banks can execute monetary policy near the short end of the maturity spectrum (see McFayden et al., 1991). Our paper takes the scant nature of the literature on developing countries into consideration by re-examining the relationship between interest rates for a developing country, Ghana. 3. Methodology 3.1 The Model Our empirical model stems from the implications of the expectation theory of yield curves. A simple representation of Eq. (1) can be formulated, following MacDonald and Speight (1988), McFayden et al. (1991), Mustafa and Rahman (1995), Arize et al. (2002), and Sarno et al. (2007) as RR tt Ω 0 Ω 1 rr tt = μμ tt (2) where RR tt denotes the long-term interest rate; rr tt denotes the short-term interest rate; Ω 0 and Ω 1 are vectors of coefficients; and μμ tt denotes a random disturbance term. The rest of the paper attempts to establish that μμ tt is integrated of order zero, I(0), and to provide the appropriate empirical representation of Eq. (2). 3.1 Asymmetric Cointegration Test Interest rates are known to contain unit roots (see Rose, 1988; Stock and Watson, 1988; Taylor et al., 2001). So, we examine the existence of unit roots in the interest rates using the Dickey-Fuller Generalized Least Squares (DF-GLS) and the Ng-Perron tests. We determine the optimal lags in the augmented Dickey-Fuller regressions of these tests using the Modified Akaike Information Criterion (MAIC). The drawback of these tests is that they assume linearity in the datagenerating process of the series under consideration. However, if the characteristic meanreverting process exhibits nonlinearities or asymmetries, then these unit roots tests will frequently fail to reject the null hypothesis of unit root (see Kapetanios et al., 2003). To overcome this problem, we utilize the Kapetanios-Shin-Snell (KSS) test developed by Kapetanios et al. (2003). These tests are well documented in the literature. Thus, we do not discuss them here due to space consideration. If the interest rates are found to contain unit roots, it is possible that they can co-move in the long run (see Engle and Granger, 1987). So, we test the possibilities of co-movements or cointegration in the interest rates using two linear cointegration techniques: (i) the Johansen technique proposed by Johansen (1988), Johansen and Juselius (1990), and Johansen (1991; 1995); and (ii) the Engle-Granger two-step technique developed by Engle and Granger (1987). 2 As with the classical unit root tests, these linear cointegration tests assume that the underlying 2 These are well-documented tests in the literature. Due to space limitation, we do not discuss them here. 5

7 variables co-move in a linear fashion. However, most macroeconomic variables including interest rates have been found to co-move nonlinearly or asymmetrically over the business cycle. For example, Granger and Lee (1989) found that sales, production, and inventories in the U.S. exhibit asymmetric adjustment toward a long-run multi-cointegrating relationship. To provide robust verification of potential cointegrating relationship between the interest rates, we use the asymmetric cointegration test develop in Enders and Siklos (2001). This asymmetric cointegration technique assumes two-regime threshold and is a simple extension of the Engle-Granger two-step technique. Enders and Siklos (2001) propose the following specification of the asymmetric adjustment mechanism qq μμ tt = ψψ 1 II tt μμ tt 1 + ψψ 2 (1 II tt )μμ tt 1 + φφ ii μμ tt ii + εε tt (3) II tt = 1 if μμ tt 1 ττ, 0 otherwise Or II tt = 1 if μμ tt 1 ττ, 0 otherwise ii=1 where II tt is the Heaviside indicator, ψψ 1, ψψ 2 and φφ ii are the coefficients, qq is the number of lags, and ττ is the threshold value. To determine qq, which accounts for the order of autocorrelated residuals, Enders and Siklos (2001) suggest we use the AIC and the BIC. The Heaviside indicator, II tt, can be specified in two ways. First, (3) and (4a) referred as the Threshold Autoregression (TAR) model; and second, (3) and (4b) referred as the Momentum Threshold Autoregression (MTAR) model. The TAR model accounts for potential nonlinear deep movements in the residual, whereas the MTAR model accounts for potential steep variability in the residual (see Enders and Granger, 1998; and Enders and Siklos, 2001). The existence of negative deepness ( ψψ 1 ψψ 2 ) implies increases are persistent, and decreases move faster to equilibrium. The MTAR model offer valuable insight when the adjustment mechanism exhibits great momentum in one direction, as opposed to the other (see Enders and Granger, 1998). The threshold value, ττ, can be specified in two ways, for the TAR and MTAR models. We can set ττ to zero (i.e. ττ = 0), leaving the names of the models intact. Or we can determine the threshold value from the data by utilizing the search method proposed by Chan (1993). Should we determine the threshold value with the search method; the resulting models become the consistent TAR and MTAR models. 3 To determine the threshold value using Chan s (1993) search method, we follow specific steps. We first sort the threshold variable, μμ tt 1 for TAR, and μμ tt 1 for MTAR, in ascending order. Then, we determine the potential threshold values. Enders (2004) recommends that the threshold value should lie between the minimum and the maximum values of the threshold variable. In practice, we must discard the lowest and highest 15% of the threshold values during the search, to allow for sufficient observations on either side of the sample. Finally, the values of the threshold variable that fall within the middle 70% band are used as potential threshold values to estimate the consistent TAR and MTAR models. 3 Chan (1993), indeed, demonstrates that the threshold value is superconsistent if it results from the minimum sum of squared errors of the fitted model after searching over the potential threshold values (see also Enders and Siklos, 2001). 6 (4a) (4b)

8 This discussion suggests that the asymmetric cointegration test is based on four models: (i) TAR with ττ = 0; (ii) consistent TAR with estimated ττ; (iii) MTAR with ττ = 0; and (iv) consistent MTAR with estimated ττ. 4 The model which best fits the dataset is the one with the minimum AIC and BIC (see Enders and Siklos, 2001). After obtaining the best model, we can test for asymmetric cointegrating relationship in the interest rates using two tests. First, the null hypothesis, HH 0 : ψψ 1 = ψψ 2 = 0, of no cointegration against the alternative of cointegration with TAR or MTAR adjustment scheme. This is a non-standard F-test with test statistic Φ and critical values reported in Enders and Siklos (2001). Second, the null hypothesis, HH 0 : ψψ 1 = ψψ 2, of linear equilibrium adjustment scheme against a nonlinear adjustment scheme alternative. This test follows a standard F-distribution. 3.2 Nonlinear Error Correction Estimation If we establish asymmetric cointegrating relationship in the interest rates or the yield curves, there is theoretical justification to fit an error correction model in the spirit of the Engle-Granger representation theorem (see Engle and Granger, 1987). In this paper, we use the asymmetric error correction model discussed in Granger and Lee (1989), Balke and Fomby (1997), and Enders and Granger (1998) to fit the interest rates. This model accounts for threshold effects and asymmetric dynamics in the underlying series of the form Δyy tt = ηη 11 + ππ + 11 EE + tt 1 + ππ 12 EE tt 1 + φφ + + jj1 Δyy tt jj JJ jj=1 JJ Δxx tt = ηη 21 + ππ + 21 EE + tt 1 + ππ 22 EE tt 1 + φφ + + jj3 Δyy tt jj jj=1 JJ + φφ jj2 Δyy tt jj jj=1 JJ + φφ jj4 Δyy tt jj jj=1 JJ + Θ + + jj1 Δxx tt jj jj=1 JJ + Θ + + jj3 Δxx tt jj jj=1 JJ + Θ jj2 Δxx tt jj jj=1 JJ + Θ jj4 Δxx tt jj jj=1 + vv tt1 (5aa) + vv tt2 (5bb) where Δyy tt and Δxx tt are the first difference of the interest rates, ηη, ππ, φφ, and Θ are coefficients to be estimated, JJ is the number of lags to be included, tt is the time subscript, and vv denotes the white-noise error term. The choice of JJ, the number of lags, is determined by the AIC and the BIC. The lagged first difference interest rates, Δyy tt jj and Δxx tt jj, are decomposed into positive and negative components. The error correction terms EE tt 1 are also decomposed into positive and negative components such that EE + tt 1 = II tt μμ tt 1 and EE tt 1 = (1 II tt )μμ tt 1 following the threshold cointegration specifications in (4), (5a), and (5b). This decomposition ensures that asymmetric shocks (both negative and positive), and threshold effects are incorporated into the error correction model (see Sun, 2011). ππ 11 and ππ 12 will be positive, and ππ 21 and ππ 22 negative, if yy tt drives the cointegrating relationship. The reverse holds, if xx tt drives the cointegrating relationship. We can investigate the existence of Granger causality by setting: (i) φφ + jj1 = φφ jj2 = 0, yy tt does not cause itself or Θ + jj1 = Θ jj2 = 0, xx tt does not cause yy tt ; and (ii) Θ + jj3 = Θ jj4 = 0, xx tt does not cause itself or φφ + jj3 = φφ jj4 = 0, yy tt does not cause xx tt. Next, we can also test the existence of distributed lag asymmetric effect by setting φφ + 11 = φφ 12 to test the hypothesis that at first lag, yy tt has 4 See Sun (2011), for similar discussion. 7

9 symmetric effect on itself; the process is replicated for each lag and for Θ + jj1 = Θ jj2 to examine the asymmetric effect of xx tt on yy tt at the jjjjh lag. In addition, we can test for the cumulative JJ + JJ symmetric effect of yy tt on itself by setting jj=1 φφ jj1 = jj=1 φφ jj2 ; the cumulative symmetric effect JJ + JJ of xx tt on yy tt by setting jj=1 Θ jj1 = jj=1 Θ jj2 ; the cumulative symmetric effect of yy tt on xx tt by JJ + JJ setting jj=1 φφ jj3 = jj=1 φφ jj4 ; and the cumulative symmetric effect of xx tt on itself by setting JJ + JJ jj=1 Θ jj3 = jj=1 Θ jj4. Finally, we can test for asymmetric equilibrium path by setting ππ 11 = ππ 12 for (5a) and ππ 21 = ππ 22 for (5b). 3.3 Data The data on the two interest rates are extracted from the Bank of Ghana s Monetary Time Series Data, available on the bank s website. We supplement missing data points for some of the months using data from the International Financial Statistics (IFS) database compiled by the IMF. We extract the monetary policy rate (MPR), and the yield on one-year government bonds (TBR) for our empirical analysis. Our choice of the Monetary Time Series Data and the IFS Data is informed by the originality of these datasets. The sample period spans January 2007 to July This period covers the Bank of Ghana s (BOG) inflation-targeting framework. The restricted sample size makes sense for two reasons. First, the techniques we use are able to account for not more than two thresholds or regimes. Second, it is more likely that the behaviour of interest rates in the current regime, the inflation-targeting regime, will be of interest to investors and policymakers than the three previous regimes Results 4.1 Summary Statistics of Monetary Policy Rate and Treasury Bill Rate Table 1 displays the key statistics of the Bank of Ghana s monetary policy rate (MPR) and the returns on one-year treasury bills (TBR) in Ghana. On the average, the MPR has been lower than the TBR over the sample period (i.e. January 2007 to July 2016). The MPR averaged approximately 16.83%, while the TBR averaged approximately 18.04% over the sample period. The TBR has been approximately 1.21% higher, on the average, than the MPR during this period. This appears to be consistent with the theory because lenders are known to demand higher interest on long-term loans due to their associated higher risks (see Campbell and Viceira, 2001; Ross et al., 2012). The theory suggests that long-term debt instruments command higher interest rates due to two main reasons. First, there is a higher probability that interest rates will increase within a longer time period than within a shorter period. This likely increase in interest rates exerts negative impact on bond prices. Second, long-term bonds generally have longer maturities than short-term bonds. Thus, a given change in interest rate will have greater impact on long-term bonds than on short-term bonds (see Campbell and Viceira, 2001; Ross et al., 2012). The maximum MPR recorded over the sample period is 26.00% which occurred between 11/30/2015 and 7/31/2016. The short-term interest rate (MPR) has been very high during this 5 For detailed classification of the monetary policy regimes in Ghana, the reader may refer to Bawumia (2010). 8

10 period, perhaps reflecting the unfavourable economic conditions in the country in recent times. The maximum TBR is 25.82% which occurred somewhere around 10/31/2014 and 12/31/2014 (see Figure 1). The high TBR during this period could be attributed to the growing government debt and souring consumer prices during this period. For example, inflation has increased about 0.5% each month from January 2014 to December 2016 (see International Financial Statistics). This incremental rate of inflation which reduces the returns from investment may have dissuaded investors from investing in treasury bills (TB), hence exerting upward pressure on the TBR. Government debt to GDP, which was in 2013, increased to by the end of 2014 an increase of about 11.96%. 6 This perhaps signaled to TB investors that the government may not be able to honour its debt obligations. Therefore, the TBR may have increased to compensate the risk-loving investors. The minimum MPR recorded during the sample period is 12.50% which occurred between 1/31/2007 and 10/31/2007, and between 7/31/2011 and 1/31/2012. For the TBR, the minimum over the sample period is 9.05% which occurred on 10/31/2011, about four years after crude oil was discovered in the country. Not surprisingly, the country began to receive huge inflow of capital from crude oil and natural gas production during This suggests that the government was not under pressure to raise funds from the sale of treasury bills, reflecting in the low returns on this debt instrument. Table 1: Descriptive Statistics of Monetary Policy Rate and Treasury Bill Rate Statistic Monetary Policy Rate Treasury Bill Rate Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Sum Sum Sq. Dev Observations Note: Std. Dev. and Sum Sq. Dev. denote standard deviation and sum of squared deviations, respectively. Figure 1 shows the yield curves for the two interest rates in Ghana from January 2007 to July These curves are sometimes known as the term structure of interest rates (see Estrella and Mishkin, 1997; Malkiel, 2015). A careful inspection suggests that these yield curves are inverted between 1/31/2007 and 1/31/2012. The fact that these curves are inverted is because the MPR exceeded the TBR at some point. Towards the late-2000s, the government of Ghana shifted the economy steadily from a controlled regime to a more market-friendly regime. The central bank 6 See for this data. 9

11 implemented interest rate policies that reflected the economic conditions prevailing in the economy at the time. In fact, the inflation-targeting framework was successfully implemented in Perhaps, the increase in the MPR during 2007 may have tightened economic conditions to the extent that investors were unable to save more for investment purpose, thereby forcing the TBR upwards between 2008 and 2009 (see Figure 1). For prolong period (i.e. between 5/31/2012 and 7/31/2015), the TBR was successively higher than the MPR. The figure shows clearly that the two interest rates are strongly correlated over time. In the next sections, we provide formal examination of this phenomenon. In particular, we examine whether there is a particular force which tends to pull these series closer to each other over time. Figure 1: The Monetary Policy Rate and the Treasury Bill Rate MPR TBR /31/2007 5/31/2007 9/30/2007 1/31/2008 5/31/2008 9/30/2008 1/31/2009 5/31/2009 9/30/2009 1/31/2010 5/31/2010 9/30/2010 1/31/2011 5/31/2011 9/30/2011 1/31/2012 5/31/2012 9/30/2012 1/31/2013 5/31/2013 9/30/2013 1/31/2014 5/31/2014 9/30/2014 1/31/2015 5/31/2015 9/30/2015 1/31/2016 5/31/2016 Source: Plotted by author using data from the Bank of Ghana s Monetary Time Series Data. 4.2 Results of Cointegration Tests We examine the possibilities of unit roots in the two interest rates, namely: MPR and TBR, using the DF-GLS, Ng-Perron, and KSS tests and find them to be first difference stationary (see Table A.1 in Appendix). So, we proceed to investigate the potential cointegrating relationship between the interest rates using linear and nonlinear cointegration tests. Table 2 reports the results of the Johansen test. The maximum eigen value statistic (λλ mmmmmm ) is reported at the upper panel, whilst the trace statistic (λλ tttttttttt ) is reported at the lower panel. The Johansen test suggests that the evidence of cointegrating relationship between the MPR and the TBR is weak at the conventional level of significance using the trace and the maximum eigen value statistics (see Table 2). In panel [1] of Table 3, we report the results of the Engle-Granger two-step test. The statistic of this residual unit root test ψψ is which is significant at 10% level. Thus, the Engle-Granger two-step test confirms that the two interest rates are cointegrated at 10% level of significance. These two tests (i.e. Engle-Granger two-step and Johansen tests) report weak 10

12 evidence of cointegration. This is particularly not surprising because these two tests assume that the MPR and the TBR are cointegrated in a linear fashion which may be untrue. Therefore, we implement the nonlinear cointegration test advanced by Enders and Siklos (2001) to account for any potential nonlinearity in the relationship between the MPR and the TBR. Table 2: Results of Johansen Cointegration Test Test Specification Lag Statistic Critical Value 10% 5% 1% Johansen λλ mmmmmm r=1 Trend r=0 Trend * r=1 Constant r=0 Constant r=1 None r=0 None * Johansen λλ tttttttttt r 1 Trend ** r=0 Trend r 1 Constant r=0 Constant ** r 1 None r=0 None Note: r is the number of cointegrating vectors; * and ** denote significance 10% and 5%, respectively. The results for the nonlinear or asymmetric cointegration test are reported in Table 3. Panels [2] to [5] show the four models that we discussed earlier. The optimal lag determined by the AIC and the BIC for the analysis is 4. We first estimate the TAR model with ττ = 0 and report the results in Panel [2] of Table 3. The point estimates of ψψ 1 = and ψψ 2 = indicates that there is convergence. Also, Φ = is greater than the critical value at 5% (i.e ). Thus, the null hypothesis of no cointegration between the monetary policy rate (MPR) and the yield on long-term government bonds (TBR) can be rejected. The adjustment mechanism is asymmetric because the p-value of under the F-statistic is smaller than (see Panel [2]). The results of the MTAR version for ττ = 0 are reported in Panel [4]. The point estimates of ψψ 1 = and ψψ 2 = implies that there is convergence. Also, Φ = is greater than the critical value at 1% (i.e ). This means we can reject the null hypothesis of no cointegration between MPR and TBR. The adjustment process to equilibrium is asymmetric, since the p-value of under the F-statistic is smaller than Next, Panels [3] and [5] display the results for the TAR and MTAR models with unknown ττ. Chan s (1993) search method is deployed to determine the best threshold. The consistent threshold for the TAR model is 0.076; whereas the consistent threshold for the MTAR model is The point estimates of ψψ 1 = and ψψ 2 = for the consistent TAR model imply that there is convergence. These estimates suggest that the speed of adjustment is faster for negative than for positive deviations. Moreover, Φ = is greater than the critical value near 5% level (i.e ), suggesting that we can reject the null hypothesis of no cointegration 11

13 between MPR and TBR. The adjustment scheme is asymmetric, since the p-value of computed under the F-statistic is smaller than (see Panel [3]). The point estimates of ψψ 1 = and ψψ 2 = for the MTAR suggests that there is convergence. That aside, Φ = is greater than the critical value near 1% level (i.e ), suggesting that we can reject the null hypothesis of no cointegration between the MPR and the TBR. The adjustment scheme is also asymmetric, since the p-value of computed under the F-statistic is smaller than (see Panel [5]). As discussed earlier, the appropriate model to proceed with is the one which yields the smallest AIC and BIC. From our analysis, the consistent TAR model yields the smallest AIC and BIC. Hence, the consistent TAR model with asymmetric equilibrium adjustment characterizes the observe nature of interest rate co-movements in Ghana. Generally, the evidence of cointegration between the interest rates confirms the expectation hypothesis of yield curves. Focusing on the consistent TAR with threshold ττ = 0.076, we find that positive deviations from the long-term equilibrium due to increases in the MPR or decreases in TBR ( μμ tt ) are corrected at 0.3% per month. Negative deviations from the long-term equilibrium due to decreases in the MPR or increases in the TBR ( μμ tt 1 < 0.076) are corrected at 8.8% per month (see Panel [3]). It is evident that negative deviations are corrected faster than positive deviation. Table: Results of Engle-Granger and Threshold Cointegration Tests Item [1] [2] [3] [4] [5] Engle-Granger TAR Consistent TAR MTAR Consistent MTAR Lag Threshold ψψ * ** *** *** * (1.684) (-2.213) (-3.057) (-2.917) (-1.659) ψψ * * * ** (-1.846) (-1.679) (-1.722) (-2.146) Total observations Cointegration obs Diagnostics AIC BIC Hypotheses Φ(HH 0 : ψψ 1 = ψψ ** 8.667** 8.492*** *** = 0) CV(1%) CV(5%) FF(HH 0 : ψψ 1 = ψψ 2 ) *** 6.348** *** 8.491*** p-value Note: a) ψψ 1 = ψψ 2 for the Engle-Granger cointegration test, so that we report only ψψ 1 = ψψ. 12

14 b) Φ is the threshold cointegration test whose critical values are reported from Tables 1 and 5 of Enders and Siklos (2001). c) FF denotes the test of asymmetric adjustment to equilibrium. d) t-statistics are reported in the parentheses. e) *, ** and *** denote significance at 10%, 5% and 1%, respectively. 4.3 Results of the Asymmetric Error Correction Model The model that best fit the observed characteristics of the yield curves in Ghana is the consistent TAR model (see Section 4.2). Since the MPR and the TBR are cointegrated with asymmetric adjustment, we proceed by fitting the corresponding error correction model. Based on the AIC and BIC, the optimal lag deemed sufficient for fitting the asymmetric error correction model is four. The asymmetric error correction model is reported as Eqs. (6a) and (6b). Note that xx tt and yy tt denote, respectively TBR and MPR. We find seven parameters to be significant in the MPR equation at the conventional levels (i.e. Θ 1 +, Θ 1, Θ 4, φφ 1 +, φφ 1, ππ + and ππ ). In the TBR equation, however, we find only four parameters to be significant (i.e. Θ 1, φφ 2 ππ +, and ππ ). The RR 2 reported for the MPR and TBR equations, respectively, are and 0.233, suggesting that the MPR equation is better specified than the TBR equation (see Table 4). Δyy tt = EE + tt EE tt Δyy Δyy Δxx Δxx Δyy 3 (6aa) [-2.161]** [-1.551]* [2.187]** [2.547]** [2.714]*** [5.755]*** [3.388]*** [-2.945]*** Δxx tt = EE + tt EE tt Δyy Δyy 0 (6bb) [2.348]** [-2.027]** [-3.119]*** [-2.069]** [2.656]*** Notes: *, ** and *** denote, respectively significance at 10%, 5% and 1%. Δ is the first difference operator. In the next few lines, we consider a series of hypotheses. H01 and H02 are the hypotheses for Granger causality between the MPR and the TBR. The F-statistic of under H01 with a p- value of suggests a causal flow from the MPR to the TBR. Similarly, the F-statistic of under H02 with a p-value of suggests a causal flow from the TBR to the MPR. However, the MPR is influenced by its lags but the TBR is not. These results indicate bidirectional causal flow between the monetary policy rate (MPR) and the returns on treasury bills in Ghana. This evidence also implies that the TBR has strong influence on the MPR during the study period, which is consistent with the forward-looking monetary policy pursued by Bank of Ghana under the inflation-targeting framework and some existing studies. In particular, our results are consistent with the liquidity premium theory which argues that the long-term interest rates exert causal influence on the short-term rates (see McFayden et al., 1991). The Bank of Ghana s Monetary Policy Committee convenes regularly to set the MPR in direct response to changing economic conditions of which the returns on one-year treasury bills form part. The returns on treasury bills, however, are influenced by market forces of which the Bank of Ghana has no influence, in principle. This evidence is similar to the evidence found for other central banks which pursue similar policies (see Enders and Siklos, 2001; Naraidoo and Paya, 2012). The remaining hypotheses concern asymmetric effects. H03 and H04 are hypotheses for distributed lag asymmetric effect. We find evidence of distributed lag asymmetric effect for both 13

15 the MPR and the TBR at the conventional levels. H05 and H06 are hypotheses of cumulative lag asymmetric effect. The F-statistic of for the TBR equation implies that cumulative lag effects from the MPR are asymmetric on the TBR. The F-statistic of for the MPR equation suggests that cumulative lag effects from the TBR are asymmetric on the MPR at conventional levels as well. However, own cumulative lag effects are symmetric for both variables. Finally, H07 is the hypothesis for asymmetric adjustment to equilibrium effect. The F-statistic for the TBR equation is with a p-value of 0.042, suggesting that the equilibrium adjustment path is asymmetric. Moreover, the F-statistic for the MPR equation is with a p-value of 0.000, suggesting that the equilibrium adjustment path is asymmetric. The implication is that, in the short term, positive deviations in the TBR are corrected at 0.34% monthly, whereas negative deviations are corrected at 8.60% monthly. Thus, the TBR responds faster to negative than positive deviations. Table 4: Results of Nonlinear Causality Test Item MPR TBR Estimate t-statistic Estimate t-statistic RR _ _ HH 01 : Θ + ii = Θ ii = 0 for all lags *** [0.000] 1.857* [0.080] HH 02 : φ + ii = φ ii = 0 for all lags 2.054** [0.048] [0.670] HH 03 : Θ + 2 = Θ *** [0.000] [0.940] HH 04 : φ + 4 = φ [0.950] 4.450** [0.038] HH 05 : Θ ii = Θ ii [0.100] 4.352** [0.040] ii 4 ii 4 + HH 06 : φφ ii = φφ ii ii ii 4.557** [0.034] [0.850] HH 07 : ππ + = ππ 7.438*** [0.000] 3.811** [0.042] Notes: a) H 01 and H 02 are tests for Granger Causality. b) H 03 and H 04 are tests for distributed lag asymmetric effect. c) H 05 and H 06 are tests for cumulative asymmetric effect. d) H 07 is the test for asymmetric adjustment to equilibrium effect. e) *, ** and *** denote significance at 10%, 5% and 1%, respectively. f) P-values are in the parentheses. 5. Concluding Remarks In this paper, we examine the nature of the yield curves in Ghana using three nonlinear econometric techniques that have been found to fit data on macroeconomic variables such as interest rates very well. These techniques are: Kapetanios et al. (2003) nonlinear unit root test; Enders and Siklos (2001) threshold cointegration test; and asymmetric error correction model with threshold adjustments documented in Granger and Lee (1989), Balke and Fomby (1997), and Enders and Granger (1998). We proceed by analyzing the stationary properties of the Bank of Ghana s monetary policy rate (MPR) and the treasury bill rate (TBR) and find them to be first difference stationary. We, then, examine whether the MPR and the TBR are cointegrated and find them to be cointegrated with threshold adjustments, which is in line with the expectation theory of yield curves. In particular, we find positive deviations from the long-term equilibrium due to increases in the MPR or decreases in the TBR to be corrected at 0.3% monthly. Negative 14

16 deviations from the long-term equilibrium due to decreases in the MPR or increases in TBR are corrected at 8.8% monthly. Since the MPR and the TBR are cointegrated with threshold adjustments, we estimate an asymmetric error correction model and test for causality between these series. Our results show causal flows in both directions between the MPR and the TBR. The causal flow from the TBR to the MPR is stronger statistically, suggesting that the TBR has strong influence on the MPR during the study period. This evidence appears to be consistent with the forward-looking monetary policy pursued by Bank of Ghana under the inflation-targeting framework and some existing studies, as well as the liquidity premium theory. We also find that, in the short term, positive deviations in the TBR are corrected at 0.34% monthly, while negative deviations are corrected at 8.60% monthly meaning that the TBR responds faster to negative than positive deviations. These findings offer another channel through which the monetary authority may pursue its policies. The knowledge that the long-term rate influences the shortterm rate means that the monetary authority can set its repo rate by not only considering the level of inflation and unemployment as it does under its inflation-targeting framework, but also by considering the yield on long-term government bonds. In other words, the BOG can execute monetary policy near the short end of the maturity spectrum. It is clear from our results that models of yield curves should incorporate nonlinearities. This may also extend to bond pricing models in general. To the market participants, our results provide an additional source of information about the future level of the short-term interest rate. Specifically, since our results are consistent with the liquidity premium theory, the spread between the long-term rate and the short-term rate will be useful in determining the future level of the short-term rate, and thereby informing their portfolio allocation decisions. The limitation of this study is that it does not forecast the future path of the MPR and the TBR which may be useful for investors. Future studies may consider using time varying regressions to forecast these interest rates. It may also be empirically useful for future studies to consider the behaviour of the yield curves under different monetary regimes in Ghana. References Aling, P., and Hassan, S. (2012). No-Arbitrage One-Factor Models of the South African Term Structure of Interest Rates. South African Journal of Economics 80(3): Arize, A. C., Malindretos, J., and Obi, Z. K. (2002). Long and Short-Term Interest Rates in 19 Countries: Test of Cointegration and Parameter Instability. Atlantic Economic Journal 30(2): Balke, N. S., and Fomby, T. (1997). Threshold Cointegration. International Economic Review 38: Bank of Ghana: Monetary Time Series Data. Available at Bawumia, M. (2010). Monetary policy and financial sector reform in Africa: Ghana's experience. Available at 15

17 Bekaert, G. and Hodrick R. J. (2001). Expectations Hypotheses Tests. Journal of Finance 56: Bernanke, B. S. (1990). On the Predictive Power of Interest Rates and Interest Rate Spreads. New England Economic Review. Nov/Dec: Bernanke, B. S. and Blinder, A. S. (1989). The Federal Funds Rate and the Channels of Monetary Transmission. Unpublished Paper. Princeton University, September. Campbell, J. Y. and Viceira, L. M. (2001). Who Should Buy Long-Term Bonds? The American Economic Review 91 (1): Chan, K. S. (1993). Consistency and Limiting Distribution of the Least Squares Estimator of a Threshold Autoregressive Model. The Annals of Statistics 21: Chen, N-F. (1991). Financial Investment Opportunities and the Macroeconomy. Journal of Finance. 46: Clarida, R. H., Sarno, L., Taylor, M. P., Valente, G., (2006). The role of asymmetries and regime shifts in the term structure of interest rates. Journal of Business 79: Cox, J. C., Ingersoll Jr., J. E., Ross, S. A. (1985). A Theory of the Term Structure of the Interest Rates. Econometrica 53: Della Corte, P., Sarno, L., and Thornton, D. L. (2008). The expectation hypothesis of the term structure of very short-term rates: Statistical tests and economic value. Journal of Financial Economics 89(1): Dube, S., and Zhou, Y. (2013). South Africa s Short Term and Long Term Interest Rates: A Threshold Cointegration Analysis. Business and Economic Research 3(1): Dzigbede, K. D., and Ofori, M. (2004). Forecasting Real Interest Rates Using the Yield Curve: Evidence from Ghana. Banking of Ghana Working Paper: WP/BOG-2004/07 Enders, W. (2004). Applied Econometric Time Series, Second Ed. Wiley, New York. Enders, W., and Granger, C. W. J. (1998). Unit-Root Tests and Asymmetric Adjustment with an Example Using the Term Structure of Interest Rates. Journal of Business & Economic Statistics 16: Enders, W., and Siklos, P. L. (2001). Cointegration and Threshold Adjustment. Journal of Business and Economic Statistics 19: Engle, R. F., and Granger, C. W. J. (1987). Cointegration and Error Correction: Representation, Estimation, and Testing. Econometrica 55:

18 Estrella, A. and Hardouvelis, G. (1991). The Term Structure as a Predictor of Real Economic Activity. The Journal of Finance, 46: Estrella, A. and Mishkin, F., The Predictive Power of the Term Structure of Interest Rates in Europe and the United States: Implications for The European Central Bank, European Economic Review, 41(7): Granger, C. W. J., and Lee, T. H. (1989). Investigation of Production, Sales, and Inventory Relationships Using Multicointegration and Non-symmetric Error-Correction Models. Journal of Applied Econometrics 4: S145 S159. Hall, A. D., Anderson, H. M., Granger, C. W. J. (1992). A Cointegration Analysis of Treasury Bill Yields. Review of Economics and Statistics 74: Harvey, C. R. (1988). The Real Term Structure and Consumption Growth. Journal of Financial Economics 22: Harvey, C. R. (1991). The Term Structure and World Economic Growth. Journal of Fixed Income 1: Harvey, C. R. (1993). The Term Structure Forecasts Economic Growth. Financial Analysts Journal (May/June): 6-8. Johansen, S. (1988). Statistical Analysis of Cointegration Vectors. Journal of Economic Dynamics and Control 12: Johansen, S. (1991). Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models. Econometrica 59: Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press. Johansen, S., and Juselius, K. (1990). Maximum Likelihood Estimation and Inference on Cointegration with Applications to the Demand for Money. Oxford Bulletin of Economics and Statistics 52(2): Kapetanios G., Shin, Y., and Snell, A. (2003). Testing for a Unit Root in the Nonlinear STAR Framework. Journal of Econometrics 112: MacDonald, R., and Speight, A. (1988). The Term Structure of Interest Rates in the UK. Bulletin of Economic Research 40: Malkiel, B. G. (2015). Term structure of interest rates: expectations and behavior patterns. Princeton University Press. 17

19 Mandeno, J., and Giles, E. A. (1995). The Expectations Theory of the Term Structure: a Cointegration/Causality Analysis of U.S. Interest Rates. Applied Financial Economics 5: McCallum, B. (1983). A Reconsideration of Sim's Evidence Concerning Monetarism. Economic Letters 13: McEnally, R. (1983). The Term Structure of Interest Rates. In Handbook of Fixed Income Securities Homewood, IL, Dow-Jones - Irwin, (F. Fabozzi and I. Pollack, eds.). McFadyen, J., Pickerill, K., and Devaney, M. (1991). The expectations hypothesis of the term structure: More Evidence. Journal of Economics and Business, 43(1), Meiselman, D. (1962). The Term Structure of Interest Rates. Engle-Wood Cliffs, NJ: Prentice- Hall. Mustafa, M., and Rahman, M. (1995). Cointegration Between U.S. Short-term and Long-term Interest Rates Both Nominal and Real. Applied Financial Economics 5: Naraidoo, R., and Paya, I. (2012). Forecasting monetary policy rules in South Africa. International Journal of Forecasting 28(2): Piazzesi, M. (2010). Affine Term Structure Models in Yacine Ait-Sahalia and Lars Hansen, Handbook of Financial Econometrics: Elsevier Edited. Richard, S. F. (1978). An Arbitrage Model of the Term Structure of Interest Rates. Journal of Financial Economics 6: Rose, A. K. (1988). Is the Real Interest Rate Stable? Journal of Finance 43: Ross, S., Westerfield, R., and Jordan, B. (2012). Fundamentals of Corporate Finance. 10th Edition, Mcgraw-Hill/Irwin Series in Finance, Insurance, and Real Estate. Sarno, L., Thornton, D, and Valente, G. (2007). The Empirical Failure of the Expectations Hypothesis of the Term Structure of Bond Yields. Journal of Financial and Quantitative Analysis 42 (1): Shiller, R. (1979). The Volatility of Long-term Interest Rates and Expectations Models of the Term Structure. Journal of Political Economy 87: Siklos, P., and Granger, C. W. J. (1997). Regime Sensitive Cointegration with an Application to Interest Rate Party. Macroeconomic Dynamics 3: Stock, J. H., and Watson, M. W. (1988). Variable Trends in Economic Time Series. Journal of Economic Perspectives. American Economic Association 2(3): Sun, C. (2011). Price Dynamics in the Import Wooden Bed Market of the United States. Forestry Policy and Economics 13:

20 Taylor, M. P. (1992). Modelling the Yield Curve. The Economic Journal 102: Taylor, M.P., Peel, D.A., and Sarno, L. (2001). Nonlinear Mean-Reversion in Real Exchange Rates: Towards A Solution to the Purchasing Power Parity Puzzles. International Economic Review 42: The International Financial Statistics (2016). Available at Van Home, J. (1978). Financial Market Rates and Flows. Englewood Cliffs, NJ: Prentice-Hall. Vasicek, O. (1977). An Equilibrium Characterization of the Term Structure. Journal of Financial Economics 5: Wallace, M. S. and Warner, J. T. (1993). The Fisher Effect and the Term Structure of Interest Rates: Test of Cointegration. Review of Economics and Statistics 75: Appendix Table A.1: Results for the Linear and Nonlinear Unit Root Tests Tests MPR Level MPR Difference TBR Level TBR Difference DF-GLS [Drift] *** *** DF-GLS [Trend] *** *** Ng-Perron [Drift] *** *** Ng-Perron [Trend] *** *** KSS NA NA Note: NA denotes non-applicable. The critical values for KSS are compared to Table 1 [Case 1] in Kapetanios et al. (2003, p. 364). *** denotes significance at 1% level, respectively. 19